This was a weird one... your code is actually basically correct, however, it appears bloomapi does not support disabling caching in the way jquery does it.

When you make the jquery call you have, the actual url becomes something like this:

http://www.bloomapi.com/api/search?offset=0&key1=last_name&op1=eq&value1=LIN&key2=first_name&op2=eq&value2=JOHN&callback=jQuery111207365460020955652_1428455335256&_=1428455335257

The callback is a jsonp construct, and the _ is a way of breaking caching. However, bloomapi appears to not like this:

jQuery111207365460020955652_1428455335256({name:ParameterError,message:_ are unknown parameters,parameters:{_:is an unknown parameter}});

To get around this, you can disable cache busting like so:

$.ajax({

    type: 'GET',

    url: 'http://www.bloomapi.com/api/search?offset=0&key1=last_name&op1=eq&value1=LIN&key2=first_name&op2=eq&value2=JOHN',

    dataType: 'jsonp',

    cache: true

}).done(function(server_data) {

    console.log(server_data)

}).fail(function() { console.log(failed) });

You will have to be careful of how else you break the cache if that's an issue; the api provider may be able to provide feedback on how to do this.

In the future, you can easily check the errors you are receiving/what you are sending using a web debugger; I used Fiddler to figure this out.